A193168 Number of simple paths from (1, 1) to (n, 3) on an n X 3 grid with king moves allowed.

1, 24, 235, 2922, 38169, 494596, 6375379, 82191766, 1059980385, 13670322763, 176299392634, 2273637717194, 29321902354841, 378149186319554, 4876791443282017, 62893416210079645, 811103334429880838, 10460373436120693109, 134901938876902858230, 1739759409373842739031

Offset: 1

Author

Matías Benzo, Jul 17 2011

Keywords

• nonn
• walk

Comments

Simple paths are self-avoiding walks.

Note: Number of paths from (1, 1) to (n, 2) on a n X 2 grid matches the sequence A086347.

Example: a(2) = 24 paths from (1, 1) to (2, 3)

1 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 3)]

2 [(1, 1), (2, 2), (1, 3), (2, 3)]

3 [(1, 1), (2, 2), (2, 1), (1, 2), (2, 3)]

4 [(1, 1), (1, 2), (2, 1), (2, 2), (1, 3), (2, 3)]

5 [(1, 1), (2, 2), (2, 1), (1, 2), (1, 3), (2, 3)]

6 [(1, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

7 [(1, 1), (1, 2), (1, 3), (2, 3)]

8 [(1, 1), (2, 2), (1, 3), (1, 2), (2, 3)]

9 [(1, 1), (2, 2), (2, 3)]

10 [(1, 1), (1, 2), (2, 3)]

11 [(1, 1), (2, 2), (1, 2), (1, 3), (2, 3)]

12 [(1, 1), (2, 1), (2, 2), (1, 3), (2, 3)]

13 [(1, 1), (2, 1), (2, 2), (1, 2), (2, 3)]

14 [(1, 1), (2, 1), (1, 2), (1, 3), (2, 2), (2, 3)]

15 [(1, 1), (1, 2), (2, 1), (2, 2), (2, 3)]

16 [(1, 1), (2, 1), (2, 2), (1, 3), (1, 2), (2, 3)]

17 [(1, 1), (2, 1), (1, 2), (2, 3)]

18 [(1, 1), (1, 2), (2, 2), (2, 3)]

19 [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3)]

20 [(1, 1), (2, 1), (2, 2), (1, 2), (1, 3), (2, 3)]

21 [(1, 1), (2, 2), (1, 2), (2, 3)]

22 [(1, 1), (2, 1), (2, 2), (2, 3)]

23 [(1, 1), (2, 1), (1, 2), (2, 2), (2, 3)]

24 [(1, 1), (2, 1), (1, 2), (2, 2), (1, 3), (2, 3)]

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Row 3 of A329118.

Cf. A086347, A140518.

Formula

Conjectures from Andrew Howroyd, Nov 05 2019: (Start)

a(n) = 12*a(n-1) + 6*a(n-2) + 50*a(n-3) + 300*a(n-4) - 194*a(n-5) - 833*a(n-6) + 352*a(n-7) + 661*a(n-8) - 84*a(n-9) - 219*a(n-10) + 72*a(n-11) + 54*a(n-12) for n > 12.

G.f.: x*(1 + 12*x - 59*x^2 - 92*x^3 + 195*x^4 + 280*x^5 + 102*x^6 - 178*x^7 - 67*x^8 + 109*x^9 - 75*x^10 - 54*x^11)/((1 + x + 9*x^2 + 24*x^3 + 9*x^4)*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)).

(End)

Extensions

Offset changed and more terms from Andrew Howroyd, Nov 05 2019